"A major responsibility of teachers is to create a learning environment in which students' use of multiple representations is encouraged, supported, and accepted by peers and adults." (NCTM, 2000, p. 139)

The job of a teacher is not to "cover" a list of mathematical concepts but rather to give students ample time and opportunities to discover the relationships between their world and the world of mathematics, and the meaning of these relationships.

Imagine a mathematics classroom in which the following occurs:

Students are expected to develop their own strategies or procedures for solving problems.

Students use representations to show their problem-solving strategies and the reasoning behind them with models, pictures, and/or words.

Students use representations to communicate their thinking to themselves, their classmates, and the teacher.

Now, this is a classroom that embraces the NCTM Standards, which is a mighty big task!

The teacher's role is to orchestrate all of these experiences for young children. Preparing a lesson involves a lot of decision making on the part of the teacher. Let's take a closer look at the Estimating Blocks lesson you saw earlier.

A problem was posed to the students: Estimate the number of blocks in the basket.

Steps

1. As a class, students looked at some models that would give them a benchmark (or referents) for making a reasonable estimate. The teacher chose to present these models to her students. In another situation, the teacher might not provide the models and would leave the problem open-ended so that students might decide to make their own benchmarks. This decision was made by the teacher based on her understanding of the needs of her students.

2. After the problem was discussed as a class, children worked in small groups to estimate the number of blocks. Conversations focused on possible strategies for making a good estimate.

3. Once the estimate had been made, the task moved to counting the actual number of blocks -- and this is where we saw the use of multiple representations among the students. Some counted one by one. Others grouped their blocks into piles of 10 to make counting easier. Some began to record their ideas on paper.

4. The teacher concentrated on the conversations and activities that were happening among the students -- and asked questions to help students make decisions if they were stuck, to encourage them to explain their thinking and ideas, and to assess where students were in their understanding of the concept.

5. Students were expected to show their ideas on paper and through the use of models. Each group of students did not go about counting the blocks in the same way. The representation of their ideas on paper differed among the groups. Again, the teacher's role was to monitor the students' thinking and to ask questions that would help them support their ideas and rethink them when appropriate.

6. After providing ample time for students to complete the task, the teacher brought them together to have them share their thinking by showing and talking about their representations. Although attention still focused on the students, their thinking, and their written record, the teacher played a critical role in orchestrating this important part of the lesson, deciding for example, which groups would share their ideas and how the discussion around those ideas would be framed.

7. The lesson was not complete with the final group presentation. Long after the students had moved on to another activity, the teacher was still considering all of the information she had gathered from watching students approach the problem, listening to their conversations, and looking at their written representations. All of this data will give the teacher the information needed to assess where her students are, what they need, and how to structure the next activity to meet those needs.

This classroom looks different from a more traditional one in which the teacher shows and tells students what to do. In a classroom where the goal is for students to become mathematically powerful, they are the ones who "show and tell" through the use of representations and communication.